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Data/DNA_EA.dat
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@ -1,5 +1,5 @@
A 73-24-5 1.572302 -1.112845488267253E-002 -1.244641965646358E-002 0.818218 -5.905035393513733E-003 -5.850193328953127E-003 0.4735 0.1725 -0.54 A 73-24-5 1.572302 -1.112845488267253E-002 -1.244641965646358E-002 0.818218 -5.905035393513733E-003 -5.850193328953127E-003 0.4735 0.1725 -0.54
C 71-30-7 1.400624 -1.104440375011201E-002 -1.232916433335030E-002 0.613380 -5.855474578578402E-003 -5.807852486273912E-003 0.2638 -0.0257 NA C 71-30-7 1.400624 -1.104440375011201E-002 -1.232916433335030E-002 0.613380 -5.855474578578402E-003 -5.807852486273912E-003 0.2638 -0.0257 NA
G 73-40-5 1.884993 -1.139399072772152E-002 -1.290878357845295E-002 1.122363 0.0 0.0 0.7487 0.4149 NA G 73-40-5 1.884993 -1.139399072772152E-002 -1.290878357845295E-002 1.122363 -6.138508297163354E-003 -6.157876784101071E-003 0.7487 0.4149 NA
T 65-71-4 1.194189 -1.125726615899225E-002 -1.278928447622424E-002 0.388406 -5.942694859945368E-003 -5.970028247300262E-003 0.0564 -0.2136 -0.29 T 65-71-4 1.194189 -1.125726615899225E-002 -1.278928447622424E-002 0.388406 -5.942694859945368E-003 -5.970028247300262E-003 0.0564 -0.2136 -0.29
U 66-22-8 1.157819 -1.116178581583812E-002 -1.263984272930559E-002 0.350665 -5.928731990069141E-003 -5.951570440565138E-003 0.0124 -0.2832 -0.22 U 66-22-8 1.157819 -1.116178581583812E-002 -1.263984272930559E-002 0.350665 -5.928731990069141E-003 -5.951570440565138E-003 0.0124 -0.2832 -0.22

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Manuscript/GW-srDFT.tex
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@ -182,7 +182,8 @@ We also compute the ionization potentials of the five canonical nucleobases (ade
\section{Introduction} \section{Introduction}
\label{sec:intro} \label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
The purpose of many-body perturbation theory (MBPT) based on Green functions is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model. \cite{MarReiCep-BOOK-16} In this approach, the \textit{screening} of the Coulomb interaction is an essential quantity that is responsible for a rich variety of phenomena that would be otherwise absent (such as quasiparticle satellites and lifetimes). \cite{Aryasetiawan_1998, Onida_2002, Reining_2017} \jt{Is it the screened Coulomb interaction which is responsible for these or more generally the Coulomb interaction? I would say the second.} The purpose of many-body perturbation theory (MBPT) based on Green functions is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model. \cite{MarReiCep-BOOK-16} In this approach, the \textit{screening} of the Coulomb interaction is an essential quantity that is responsible for a rich variety of phenomena that would be otherwise absent (such as quasiparticle satellites and lifetimes). \cite{Aryasetiawan_1998, Onida_2002, Reining_2017}
%\jt{Is it the screened Coulomb interaction which is responsible for these or more generally the Coulomb interaction? I would say the second.}
The so-called {\GW} approximation is the workhorse of MBPT and has a long and successful history in the calculation of the electronic structure of solids. \cite{Aryasetiawan_1998, Onida_2002, Reining_2017} The so-called {\GW} approximation is the workhorse of MBPT and has a long and successful history in the calculation of the electronic structure of solids. \cite{Aryasetiawan_1998, Onida_2002, Reining_2017}
{\GW} is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2015, Bruneval_2016, Bruneval_2016a, Boulanger_2014, Blase_2016, Li_2017, Hung_2016, Hung_2017, vanSetten_2015, vanSetten_2018, Ou_2016, Ou_2018, Faber_2014} thanks to efficient implementation relying on plane waves \cite{Marini_2009, Deslippe_2012, Maggio_2017} or local basis functions. \cite{Blase_2011, Blase_2018, Bruneval_2016, vanSetten_2013, Kaplan_2015, Kaplan_2016, Krause_2017, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b} {\GW} is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2015, Bruneval_2016, Bruneval_2016a, Boulanger_2014, Blase_2016, Li_2017, Hung_2016, Hung_2017, vanSetten_2015, vanSetten_2018, Ou_2016, Ou_2018, Faber_2014} thanks to efficient implementation relying on plane waves \cite{Marini_2009, Deslippe_2012, Maggio_2017} or local basis functions. \cite{Blase_2011, Blase_2018, Bruneval_2016, vanSetten_2013, Kaplan_2015, Kaplan_2016, Krause_2017, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b}
@ -336,9 +337,8 @@ For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\
In this subsection, we provide the minimal set of equations required to describe {\GOWO}. In this subsection, we provide the minimal set of equations required to describe {\GOWO}.
More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}. More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
For sake of generality, we consider a $\KS$ reference. For sake of generality, we consider closed-shell systems with a $\KS$ reference.
The one-electron energies $\e{p}$ and their corresponding (real-valued) orbitals $\MO{p}(\br{})$ (which defines the basis set $\Bas$) are then $\KS$ energies and orbitals. The one-electron energies $\e{p}$ and their corresponding (real-valued) orbitals $\MO{p}(\br{})$ (which defines the basis set $\Bas$) are then $\KS$ energies and orbitals.
\jt{equations are for closed-shell systems}
Within the {\GW} approximation, the correlation part of the self-energy reads Within the {\GW} approximation, the correlation part of the self-energy reads
\begin{equation} \begin{equation}
@ -347,15 +347,15 @@ Within the {\GW} approximation, the correlation part of the self-energy reads
\Sig{\text{c},p}{\Bas}(\omega) \Sig{\text{c},p}{\Bas}(\omega)
& = \mel*{\MO{p}}{\Sig{\text{c}}{\Bas}(\omega)}{\MO{p}} & = \mel*{\MO{p}}{\Sig{\text{c}}{\Bas}(\omega)}{\MO{p}}
\\ \\
& = 2 \sum_{i}^\text{occ} \sum_{x} \frac{[pi|x]^2}{\omega - \e{i} + \Om{x} - i \eta} & = 2 \sum_{i}^\text{occ} \sum_{m} \frac{[pi|m]^2}{\omega - \e{i} + \Om{m} - i \eta}
\\ \\
& + 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa|x]^2}{\omega - \e{a} - \Om{x} + i \eta}, & + 2 \sum_{a}^\text{virt} \sum_{m} \frac{[pa|m]^2}{\omega - \e{a} - \Om{m} + i \eta},
\end{split} \end{split}
\end{equation} \end{equation}
\jt{x should be defined. If it is just an index for the eigenvector/eigenvalue, maybe we should call it $m$ instead?} where $\eta$ is a positive infinitesimal. where $m$ corresponds to a sum over the single excitations and $\eta$ is a positive infinitesimal.
The screened two-electron integrals The screened two-electron integrals
\begin{equation} \begin{equation}
[pq|x] = \sum_{ia} (pq|ia) (\bX+\bY)_{ia}^{x} [pq|m] = \sum_{ia} (pq|ia) (\bX+\bY)_{ia}^{m}
\end{equation} \end{equation}
are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994} are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994}
\begin{equation} \begin{equation}
@ -382,11 +382,10 @@ and the transition densities $(\bX+\bY)_{ia}^{x}$ originating from a (direct) ra
with with
\begin{align} \begin{align}
\label{eq:RPA} \label{eq:RPA}
A_{ia,jb} & = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2 (ia|jb), A_{ia,jb} & = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2 (ib|aj),
& &
B_{ia,jb} & = 2 (ia|bj), B_{ia,jb} & = 2 (ia|jb),
\end{align} \end{align}
\jt{the integral in the B term could be written as in the A term since we use real-valued orbitals}
and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook} and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{x}$ which represent the poles of the screened Coulomb potential $\W{}{}$. Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{x}$ which represent the poles of the screened Coulomb potential $\W{}{}$.
@ -395,7 +394,6 @@ The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of t
\label{eq:QP-G0W0} \label{eq:QP-G0W0}
\omega = \e{p} - \Pot{\xc,p}{\Bas} + \Sig{\text{x},p}{\Bas} + \Re[\Sig{\text{c},p}{\Bas}(\omega)]. \omega = \e{p} - \Pot{\xc,p}{\Bas} + \Sig{\text{x},p}{\Bas} + \Re[\Sig{\text{c},p}{\Bas}(\omega)].
\end{equation} \end{equation}
\jt{an equation involving $Z_p$ seems to be missing!}
with the largest renormalization weight (or factor) with the largest renormalization weight (or factor)
\begin{equation} \begin{equation}
\label{eq:Z} \label{eq:Z}
@ -586,7 +584,7 @@ In a nutshell, the present basis set correction provides cc-pVQZ quality results
Besides, it allows to reach chemical accuracy with the quadruple-$\zeta$ basis set, an accuracy that could not be reached even with the cc-pV5Z basis set for the conventional calculations. Besides, it allows to reach chemical accuracy with the quadruple-$\zeta$ basis set, an accuracy that could not be reached even with the cc-pV5Z basis set for the conventional calculations.
Very similar conclusions are drawn at the {\GOWO}@{\PBEO} level (see Table \ref{tab:GW20_PBE0}) with a slightly faster convergence to the CBS limit. Very similar conclusions are drawn at the {\GOWO}@{\PBEO} level (see Table \ref{tab:GW20_PBE0}) with a slightly faster convergence to the CBS limit.
For example, at the {\GOWO}@PBE0+srLDA/cc-pVQZ level, the MAD is only $0.02$ eV with a maximum error as small as $0.09$ eV. \jt{these numbers do not exactly correspond to the ones in the Table II. Is it because of round off?} For example, at the {\GOWO}@PBE0+srLDA/cc-pVQZ level, the MAD is only $0.02$ eV with a maximum error as small as $0.09$ eV.
It is worth pointing out that, for ground-state properties such as atomization and correlation energies, the density-based correction brought a more significant basis set reduction. It is worth pointing out that, for ground-state properties such as atomization and correlation energies, the density-based correction brought a more significant basis set reduction.
For example, we evidenced in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} that quintuple-$\zeta$ quality atomization and correlation energies are recovered with triple-$\zeta$ basis sets. For example, we evidenced in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} that quintuple-$\zeta$ quality atomization and correlation energies are recovered with triple-$\zeta$ basis sets.

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Notebooks/GW100.nb
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